Connected sum

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In mathematics, specifically in topology, the operation of connected sum is a geometric modification on manifolds. Its effect is to join two given manifolds together near a chosen point on each. This construction plays a key role in the classification of closed surfaces.

More generally, one can also join manifolds together along identical submanifolds; this generalization is often called the fiber sum. There is also a closely related notion of a connected sum on knots, called the knot sum or composition of knots.

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[edit] Connected sum at a point

A connected sum of two m-dimensional manifolds is a manifold formed by deleting a ball inside each manifold and gluing together the resulting boundary spheres.

If both manifolds are oriented, there is a unique connected sum defined by having the gluing map reverse orientation. Although the construction uses the choice of the balls, the result is unique up to homeomorphism. One can also make this operation work in the smooth category, and then the result is unique up to diffeomorphism. There are subtle problems in the smooth case: not every diffeomorphism between the boundaries of the spheres gives the same composite manifold, even if the orientations are chosen correctly. For example, Milnor showed that two 7-cells can be glued along their boundary so that the result is an exotic sphere homeomorphic but not diffeomorphic to a 7-sphere. However there is a canonical way to choose the gluing which gives a unique well defined connected sum. This uniqueness depends crucially on the annulus theorem, which is not at all obvious.

The operation of connected sum is denoted by \#; for example A \# B denotes the connected sum of A and B.

The operation of connected sum has the sphere Sm as an identity; that is, M \# S^m is homeomorphic (or diffeomorphic) to M.

The classification of closed surfaces, a foundational and historically significant result in topology, states that any closed surface can be expressed as the connected sum of a sphere with some number g of tori and some number k of real projective planes.

[edit] Connected sum along a submanifold

Let M1 and M2 be two smooth, oriented manifolds of equal dimension and V a smooth, closed, oriented manifold, embedded as a submanifold into both M1 and M2. Suppose furthermore that there exists an isomorphism of normal bundles

\psi: N_{M_1} V \to N_{M_2} V

that reverses the orientation on each fiber. Then ψ induces an orientation-preserving diffeomorphism

N_1 \setminus V \cong N_{M_1} V \setminus V \to N_{M_2} V \setminus V \to N_{M_2} V \setminus V \cong N_2 \setminus V,

where each normal bundle N_{M_i} V is diffeomorphically identified with a neighborhood Ni of V in Mi, and the map

N_{M_2} V \setminus V \to N_{M_2} V \setminus V

is the orientation-reversing diffeomorphic involution

v \mapsto v / |v|^2

on normal vectors. The connected sum of M1 and M2 along V is then the space

(M_1 \setminus V) \bigcup_{N_1 \setminus V = N_2 \setminus V} (M_2 \setminus V)

obtained by gluing the deleted neighborhoods together by the orientation-preserving diffeomorphism. The sum is often denoted

(M_1, V) \# (M_2, V).

Its diffeomorphism type depends on the choice of the two embeddings of V and on the choice of ψ.

Loosely speaking, each normal fiber of the submanifold V contains a single point of V, and the connected sum along V is simply the connected sum as described in the preceding section, performed along each fiber. For this reason, the connected sum along V is often called the fiber sum.

The special case of V a point recovers the connected sum of the preceding section.

[edit] Connected sum along a codimension-two submanifold

Another important special case occurs when the dimension of V is two less than that of the Mi. Then the isomorphism ψ of normal bundles exists whenever their Euler classes are opposite:

e(N_{M_1} V) = -e(N_{M_2} V).

Furthermore, in this case the structure group of the normal bundles is the circle group SO(2); it follows that the choice of embeddings can be canonically identified with the group of homotopy classes of maps from V to the circle, which in turn equals the first integral cohomology group H1(V). So the diffeomorphism type of the sum depends on the choice of ψ and a choice of element from H1(V).

A connected sum along a codimension-two V can also be carried out in the category of symplectic manifolds; this elaboration is called the symplectic sum.

[edit] Local operation

The connected sum is a local operation on manifolds, meaning that it alters the summands only in a neighborhood of V. This implies, for example, that the sum can be carried out on a single manifold M containing two disjoint copies of V, with the effect of gluing M to itself. For example, the connected sum of a two-sphere at two distinct points of the sphere produces the two-torus.

[edit] Connected sum of knots

There is a closely related notion of the connected sum of two knots. In fact, if one regards a knot merely as a one-manifold, then the connected sum of two knots is just their connected sum as a one-dimensional manifold. However, the essential property of a knot is not its manifold structure (all knots are circles) but rather its embedding into the ambient space. So the connected sum of knots has a more elaborate definition that produces a well-defined embedding, as follows.

Consider a planar projection of each knot and suppose these projections are disjoint.
Consider a planar projection of each knot and suppose these projections are disjoint.
Find a rectangle in the plane where one pair of sides are arcs along each knot but is otherwise disjoint from the knots.
Find a rectangle in the plane where one pair of sides are arcs along each knot but is otherwise disjoint from the knots.
Now join the two knots together by deleting these arcs from the knots and adding the arcs that form the other pair of sides of the rectangle.
Now join the two knots together by deleting these arcs from the knots and adding the arcs that form the other pair of sides of the rectangle.

This procedure results in the projection of a new knot, the connected sum (or knot sum, or composition) of the original knots.

Under this operation, knots in 3-space form a commutative monoid with prime factorization, which allows us to define what is meant by a prime knot. Proof of commutativity can be seen by letting one summand shrink until it is very small and then pulling it along the other knot. The unknot is the unit. The trefoil knot is the simplest prime knot. Higher dimensional knots can be added by splicing the n-spheres.

In three dimensions, the unknot cannot be written as the sum of two non-trivial knots. This fact follows from additivity of knot genus; another proof relies on an infinite construction sometimes called the Mazur swindle. In higher dimensions, it is possible to get an unknot by adding two nontrivial knots.

[edit] See also

[edit] References

  • Robert Gompf: A new construction of symplectic manifolds, Annals of Mathematics 142 (1995), 527-595
  • William S. Massey, A Basic Course in Algebraic Topology, Springer-Verlag, 1991. ISBN 0-387-97430-X.